Math 25 Activity 11: Radicals
This last activity goes over simplifying radicals. First we want to discuss the word radical and its
connection to the symbol
.
1. What is the difference between a radical and a square root?
The term “radical” refers to the symbol
that means square root when it does not have an index
on it. When it does have an index on it, it could be a cube root
root
n
. These are all radicals. Technically, the square root
3
, a fourth root
4
could have an index
, or an n-th
2
but since it
is the most commonly used radical, we don’t write the index for square root.
Radicals use the index to ask a question about what is inside of the radical. 25 asks the question
“What number when squared is 25?” Another way of saying it is “What number times itself is 25?”
So the answer to the question is the answer when we are asked to simplify
is 5 because 5 times itself is 25.
25 , which in this case,
Other indices refer to higher exponents. For 4 81 , the question is “What number when raised to
the 4th power is 81?” Another way of saying it is “What number multiplied by itself 4 times is 81?”
The answer is 3, because 3  3  3  3  81 .
2. Take the following expressions and write the English question that is being asked. Then answer the
question.
Radical Expression
English Question Being Asked
Answer to the Question
121
3
125
4
16
All of the previous examples had counting number answers. When we have a counting number
answer, we don’t need to write the radical anymore. It is nice when the answer works out perfectly,
but there are infinitely many cases in which the radical will not be asking a question in which the
answer is a counting number. These cases are irrational numbers.
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Let’s explore some of these cases with our calculators and a number line.
3. Take the following expressions and use your calculator to find their decimal equivalent rounded to
the nearest tenth, then plot them on the number line provided.
Radical
Expression
Decimal Equivalent Rounded
to the Nearest Tenth
25
29
33
36
So when we ask the question “What number times itself is 29?” and we can’t answer a counting
number, the best approximation begins with the fact that 29 sits between the squares 25 and 36,
and so the answer to the question is somewhere between 5 and 6.
4. Without using the calculator and using your information from problem 3, what decimal is a good
estimation of
32 ?
Should we consider rounding to the nearest thousandth to be more precise considering
32 and
33 ? Why or Why not?
We need to mention here that rounding at all makes the number an approximation. Since irrational
numbers have infinitely many digits after the decimal without a repeating pattern, stopping at any
point makes the number no longer exact. There are times in which mathematicians do not want to
work with approximations, preferring exact answers. This situation is why we learn how to simplify
radical expressions and leave them in exact form.
Notice how 25 can be written in prime factorization form 5  5 . That makes it easy to answer
the original question “What number times itself is 25?” because we can see that there are two 5’s
multiplying together to make the 25 so the answer is 5. Similarly,
36 is written 2  2  3  3 which
could be thought of as 2  2  3  3 so the answer is 2  3 which is 6. This method also works with
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Instructor!
higher indices. 3 1000 is asking the question “What number times itself three times is 1000?” We
can write it using its prime factorization and look for repetitions where 3 numbers are the same,
because this is a cube root. 3 1000 becomes
which is 10.
3
2  2  2  3 5  5  5 which means the answer is 2  5
5. Take the following expressions and write the number under the radical in its prime factorization,
then write the expression in simplified form.
Radical Expression
Prime Factorization
Simplified Answer
441
3
216
4
10000
Your instructor will have groups do these problems on the board showing their prime factorizations
and their final answers.
6. What do you think happens when we look at the primes under a radical, and they don’t have the
right number of repetitions, that is, matching how many times the number should repeat as
indicated in the index?
Let’s look at radicals that don’t have a counting number answers, but we want to leave them in
simplified exact form. What that means is, we have considered all the possible numbers that could
be taken outside of the radical from a prime factorization, but some might still be left because they
can’t help us answer our question.
Example 300 asks “What number times itself gives you 300?” in which case there is no counting
number that answers this question. So we take a look at the prime factorization of 300:
2  2  5  5  3 which can be thought of as
2  2  5  5  3 . From before we see that
2  2 is just
2, which helps us answer the question, and 5  5 is just 5 which also helps us answer the question.
The 3 is the only number that doesn’t have a repeat and isn’t helping us answer the question. That
leaves us with 2  5  3 which is 10 3 . Since there is only one 3 in the prime factorization, that 3 is
stuck under the radical. We leave the answer as 10 3 which is the simplified exact answer.
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Instructor!
7. Take the following expressions and write the number under the radical in its prime factorization,
then write the expression in simplified exact form.
Radical
Expression
Prime Factorization
Simplified Exact Answer
98
360
210
3
120
Your instructor will have groups do these problems on the board showing their prime factorizations
and their final answers.
8. What do you think happens when we add variables to these expressions?
From previous activities, we have said that when working with primes, we can treat different
variables like prime numbers.
9. Take the following expressions and write the number under the radical in its prime factorization,
then write the expression in simplified exact form. Assume all variables are positive real numbers.
Radical
Expression
Prime Factorization
Simplified Exact Answer
12x 2
32 y 3
50xz 6
3
125x 3
Your instructor will have groups do these problems on the board showing their prime factorizations
and their final answers.
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